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Spectrum and L-spectrum of the power graph and its main supergraph for certain finite groups

Hamzeh Asma (Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, I.R. Iran)
Ashrafi Ali Reza (Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, I.R. Iran)

Let G be a finite group. The power graph P(G) and its main supergraph S(G) are two simple graphs with the same vertex set G. Two elements x,y  G are adjacent in the power graph if and only if one is a power of the other. They are joined in S(G) if and only if o(x)|o(y) or o(y)|o(x). The aim of this paper is to compute the characteristic polynomial of these graph for certain finite groups. As a consequence, the spectrum and Laplacian spectrum of these graphs for dihedral, semi-dihedral, cyclic and dicyclic groups were computed.

Keywords: Power graph, main supergraph, spectrum, Laplacian spectrum